Evaluating Powers of 𝒊
𝑖1 = 𝑖
𝑖 2 = −1
See the pattern:
𝑖 3 = 𝑖 2 ∗ 𝑖 1 = −1 ∗ 𝑖 = −𝑖
𝑖
𝑖 4 = 𝑖 2 ∗ 𝑖 2 = −1 ∗ −1 = 1
−1
𝑖 5 = 𝑖 2 ∗ 𝑖 2 ∗ 𝑖 1 = −1 ∗ −1 ∗ 𝑖 = 𝑖
−𝑖
𝑖 6 = 𝑖 2 ∗ 𝑖 2 ∗ 𝑖 2 = −1 ∗ −1 ∗ −1 = −1
1
𝑖 7 = 𝑖 2 ∗ 𝑖 2 ∗ 𝑖 1 = −1 ∗ −1 ∗ 𝑖 = −𝑖
𝑖 8 = 𝑖 2 ∗ 𝑖 2 ∗ 𝑖 2 ∗ 𝑖 2 = −1 ∗ −1 ∗ −1 ∗ −1 = 1
Simplify.
𝑖 42 = (𝑖 2 )21 = (−1)21 = −1
𝑖 21 = (𝑖 2 )10 ∗ 𝑖 = (−1)10 ∗ 𝑖 = 1 ∗ 𝑖 = 𝑖
𝑖 83 = (𝑖 2 )41 ∗ 𝑖 = (−1)41 ∗ 𝑖 = −1 ∗ 𝑖 = −𝑖
Imaginary Numbers
√−1 = 𝑖
2
(√−1) = 𝑖 2
−1 = 𝑖 2
Simplify.
√−16 = √16 ∗ √−1 = 4𝑖
√−20 = √20 ∗ √−1 = 2√5 ∗ 𝑖 = 2𝑖√5
(4𝑖)2 = 4𝑖 ∗ 4𝑖 = 16𝑖 2 = 16 ∗ −1 = −16
√−9 ∗ √−25 = 3𝑖 ∗ 5𝑖 = 15𝑖 2 = 15 ∗ −1 = −15
Always factor out a √−1 , which is equal
to 𝑖 , then simplify.
Adding and Subtracting Complex Numbers
(−3 + 5𝑖) + (−6𝑖 − 8)
Nothing to distribute for addition.
For subtraction do not forget to
change every sign past the minus sign
between sets of parenthesis.
Then, just combine like terms. Real
with real, and Imaginary with
imaginary.
(2 + 3𝑖) − (3 + 5𝑖)
Multiplying Complex Numbers
2𝑖(3 − 5𝑖)
Distribute.
FOIL (first, outer, inner, last)
Don’t forget: 𝑖 2 = −1
(3 + 2𝑖)(3 − 4𝑖)
Dividing Complex Numbers
3−𝑖
2−𝑖
3+8𝑖
−𝑖
Multiply by the
conjugate: (a + bi)
Multiply by the
conjugate: (+i)
Conjugate: a binomial formed by making the second term of a binomial negative.